Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group

TL;DR

This paper explores the connections between Fourier multipliers and Schur multipliers in noncommutative spaces of discrete groups, providing new insights and applications in harmonic analysis and operator theory.

Contribution

It establishes a transfer principle linking Fourier and Schur multipliers and applies it to various problems in noncommutative harmonic analysis.

Findings

Lacunary sets characterized via multipliers

Unconditional Schauder bases constructed for spectral subspaces

Norm estimates for Hilbert transform and Riesz projection on Schatten classes

Abstract

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets; unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum, that is, by a subset of the group; the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2; the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.